After reading Alan Greenspan's essay on the gold standard in Capitalism: The Unknown Ideal, I am curious to know how exactly currency (not wealth) would be handled. In the essay, he writes about banks keeping gold reserves, saying:

A free banking system based on gold is able to extend credit and thus to create bank notes (currency) and deposits, according to the production requirements of their economy.

Does this mean, under Laissez-Faire, that banks and not governments would print currency? Or would governments simply print currency according to the needs of banks? Mr. Greenspan also mentions the national currencies of a few nations as well.

Even though the units of exchange (the dollar, the pound, the franc, etc.) differ from country to country, when all are defined in terms of gold the economies of the different countries act as one...

I think that this would suggest some government minting function, but I am unsure.

Under Laissez-Faire, banks and not governments would print currency. As Greenspan points out, this is still nominally the case in the US even after the establishment of the Federal Reserve System in 1913. (The Federal Reserve banks are banks which are "government sponsored, controlled, and supported".) Under a full-reserve system this would be fine.

The essay that you reference has an interesting take on how these private banks could maintain a fractional-reserve system without everything falling apart. I don't agree with it, and in fact I don't think Greenspan himself would agree with it.

The pure laissez-faire approach to currency would entail the denationalization of money. Strictly speaking it isn't even necessary for governments to define currencies, only units of weight and measure. Money is then a fixed weight of some commodity or commodities (gold and silver being common historical examples). There's a reason the British pound sterling is called what it is -- originally it was a literal pound of literal sterling silver. Allowing governments to define units of money simply opens the door to legal debasement and introduces distortions into the money system by enforcing a fixed ratio between a measure of weight and a measure of value.

I see no need for a government minting function. It isn't like government has shown itself a responsible guardian of the currency. There is also an important feedback loop in private minting that isn't present in government minting: if the creation of money is left in private hands, new money will only be created when it is profitable to do so. Government mints, by definition, are not acting in pursuit of profit and are thus likely to produce more money than is economically required.